Directions: For a 5-digit number, without repetition of digits, the following information is available. (i) The first digit is more than 5 times the last digit. (ii) The two-digit number formed by the last two digits is the product of two prime numbers. (iii) The first three digits are all odd. (iv) The number does not contain the digits 3 or 0 and the first digit is also the largest. The second digit of the number is

### Concept & Logic This problem is solved using logical deduction based on digit properties and constraints. By systematically applying the given conditions (odd digits, inequalities, and prime factorization), we can determine the exact digits of the number. ### Step-by-Step Solution Let the 5-digit number be represented as $A B C D E$. We know there is no repetition of digits. From statement (iv), the digits $3$ and $0$ are not allowed. From statement (iii), the first three digits ($A, B, C$) are all odd. The available odd digits (excluding $3$) are $1, 5, 7, 9$. From statement (i), $A > 5E$. Since $A$ is a single digit and must be odd ($\le 9$), the maximum possible value for $A$ is $9$. Substituting $9$ into the inequality: $9 > 5E \implies E < 1.8$. Since $0$ is not allowed, $E$ must be $1$. Since $E = 1$, and $A, B, C$ must be distinct odd digits, the only remaining odd digits for $A, B, C$ are $5, 7, 9$. Therefore, $\{A, B, C\} = \{5, 7, 9\}$. From statement (iv), $A$ is the largest digit in the entire number. Thus, $A = 9$. This means the digits $B$ and $C$ must be $5$ and $7$. However, the problem does not provide any conditions to distinguish their exact order. $B$ could be $5$ and $C$ could be $7$, or vice versa. ### Exam Strategy & Shortcut In constraint satisfaction puzzles, immediately identify the most restrictive condition. Here, $A > 5E$ immediately locks $E=1$ because $A$ cannot exceed $9$. Once you deduce that $A, B, C$ must pull from $\{5, 7, 9\}$ and $A=9$, you can immediately see $B$ and $C$ are interchangeable. You don't even need to figure out $D$ to answer this specific question, saving valuable exam time. ### Common Pitfall Students often assume an ascending or descending order when constraints aren't explicitly given. Because $B$ and $C$ are not definitively ordered by the clues, assuming one specific configuration leads to an incorrect guess. Always check if a unique configuration is guaranteed. ### Final Answer Therefore, the correct answer is Cannot be determined.